The basic idea behind a linear public good game is as follows: You have a group of people, typically four in the lab, who are endowed with a certain of money, say $5. Each group member is independently asked how much they want to contribute towards a public good. Any dollar that is contributed results in everyone in the group getting a return of, say, $0.40.
From an individual's perspective contributing towards the group looks like a bad deal because you contribute $1 and only get back $0.40. Note, however, that from the group's perspective a contribution of $1 results in a total return of 4 x 0.4 = $1.60. So, from the group's perspective it is good to contribute. For instance, if all four group members contribute $5 then each gets 4 x 5 x 0.4 = $8. And $8 is better than $5.
'Standard economic theory' gives a very simple prediction in a linear public good game. Namely, the game has a unique Nash equilibrium in which everyone should contribute $0. This result follows from the basic logic that is not in a person's material interest to give $1 and get back $0.40. But, what actually happens in the lab? Well many people do contribute. Indeed the absurdity of the idea we should all contribute zero is captured in the famous Monty Python sketch about a merchant banker.
So, what happens if we try to capture people's desire to contribute for the good of the group? This depends on why people contribute. To illustrate, suppose that there are two types of individual - selfish and altruist. The selfish do best to contribute zero and maximize own payoff while the altruists do best to contribute $5 and maximize group payoff. In this case there is still a unique Nash equilibrium but now with positive contributions. In particular, the selfish contribute $0 and the altruists contribute $5. Exact contributions will depend on the number of altruists in the group.
Altruists, however, seem pretty rare in the lab. What we typically observe are conditional co-operators. These are people who will contribute if others do. Suppose that everyone in the group is a conditional co-operator. Then it is still a Nash equilibrium for everyone to contribute $0 - if nobody else contributes then I don't want to either. But, it is also a Nash equilibrium for everyone to contribute $5 - if others contribute $5 then I am willing to as well. So, we get multiple equilibria, including ones with positive contributions. In this case, exact contributions will depend on which equilibrium the group manages to coordinate on.
Finally, consider a more realistic setting in which we have a mix of selfish, altruists, conditional co-operators and everything in between. Then we have to work through what outcomes are Nash equilibria are which are not. For instance, in a group with one selfish person and three conditional co-operators, are the conditional co-operators willing to contribute and 'put up' with the selfish person in their midst?
In a recent paper on 'What are the equilibria in public-good experiments?' Irenaeus Wolff gives us some answers to this type of question. In order to do so, experimental subjects were asked to fill in a complete contribution table which says how much they would want to contribute as a function of what others are contributing. You can then group people together and work out what contributions would be consistent with Nash equilibrium.
A somewhat depressing result is that a small proportion of selfish people can result in zero contributions being the unique Nash equilibrium. For instance, in one treatment only 23% of subjects were classified as selfish and yet in 60% of groups the unique Nash equilibrium would be zero contributions. So, you only need one 'bad egg'. There is, however, a more optimistic side to this in that 40% of groups should be able to do better than zero contributions. Indeed, in the two other treatments considered the proportion of groups that can do better than zero contributions is 60 and 70%.
To say that groups can do better than zero contributions is, however, different to saying they will do better. In the lab we often see groups converge on zero contributions. Not only, therefore, do we need enough conditional co-operators in the group we also need them to somehow coordinate on a good outcome. The option of punishing those who free-ride is one way that has been shown to work.